On the asymptotic properties of solutions to a differential equation in a case of bifurcation without eigenvalues

1986 ◽  
Vol 104 (1-2) ◽  
pp. 137-159 ◽  
Author(s):  
R. J. Magnus
Keyword(s):  
Differential Equation ◽  
Euclidean Space ◽  
Linear Equation ◽  
Trivial Solution ◽  
Asymptotic Properties ◽  
Dimensional Euclidean Space ◽  
Bifurcation Problem ◽  
Asymptotic Dependence ◽  
Fredholm Type

SynopsisThe semi-linear equation Δu − λu + h(x)uσ = 0 is studied on all of d-dimensional Euclidean space. In the bifurcation problem a non-trivial solution is sought for small λ which tends to zero with λ. The asymptotic dependence of the solution on λ is examined. For fixed λ = 1 the existence of non-degenerate non-trivial solutions is proved for generic measurable h(x) sufficiently near to a constant, provided d = 1 or 3. The two problems are seen to be interdependent. The bifurcation problem at λ = 0 is particularly interesting as the linearised equation is of non-Fredholm type.

scholarly journals Flows equivalences

2001 ◽  
Vol 2 (1) ◽  
pp. 113 ◽  
Author(s):  
Gabriel Soler López
Keyword(s):  
Differential Equation ◽  
Euclidean Space ◽  
Real Line ◽  
Dimensional Euclidean Space ◽  
Open Set ◽  
Entire Real Line

<p>Given a differential equation on an open set O of an n-manifold we can associate to it a pseudo-flow, that is, a flow whose trajectories may not be defined in the entire real line. In this paper we prove that this pseudo-flow is always equivalent to a flow with its trajectories defined in all R. This result extends a similar result of Vinograd stated in the n-dimensional Euclidean space.</p>

Expansions for a fundamental solution of Laplace's equation on ℝ3 in 5-cyclidic harmonics

2014 ◽  
Vol 12 (06) ◽  
pp. 613-633
Author(s):  
Howard S. Cohl ◽  
Hans Volkmer
Keyword(s):  
Differential Equation ◽  
Euclidean Space ◽  
Fundamental Solution ◽  
Three Dimensional ◽  
Singular Points ◽  
Dimensional Euclidean Space ◽  
Laplace’S Equation ◽  
Eigenfunction Expansions ◽  
Laplace's Equation

We derive eigenfunction expansions for a fundamental solution of Laplace's equation in three-dimensional Euclidean space in 5-cyclidic coordinates. There are three such expansions in terms of internal and external 5-cyclidic harmonics of first, second and third kind. The internal and external 5-cyclidic harmonics are expressed by solutions of a Fuchsian differential equation with five regular singular points.

Oscillation of Elliptic Equations in General Domains

1975 ◽  
Vol 27 (6) ◽  
pp. 1239-1245 ◽  
Author(s):  
E. S. Noussair
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Operator ◽  
Euclidean Space ◽  
Unbounded Domain ◽  
Elliptic Equations ◽  
General Type ◽  
Elliptic Partial Differential Equation ◽  
Dimensional Euclidean Space ◽  
Image Position

Oscillation criteria will be obtained for the linear elliptic partial differential equationin an unbounded domain G of general type in n-dimensional Euclidean space En. The differential operator D is defined as usual by where each α (i), i = 1, … , n, is a non-negative integer.

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

Some properties of Wigner coefficients and hyperspherical harmonics

1956 ◽  
Vol 52 (3) ◽  
pp. 424-430 ◽  
Author(s):  
A. P. Stone
Keyword(s):  
Angular Momentum ◽  
Euclidean Space ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Hyperspherical Harmonics ◽  
Shift Operators ◽  
Analytical Relations

ABSTRACTGeneral shift operators for angular momentum are obtained and applied to find closed expressions for some Wigner coefficients occurring in a transformation between two equivalent representations of the four-dimensional rotation group. The transformation gives rise to analytical relations between hyperspherical harmonics in a four-dimensional Euclidean space.

On inequalities of the Tchebychev type

1963 ◽  
Vol 59 (1) ◽  
pp. 135-146 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Probability Theory ◽  
Euclidean Space ◽  
Random Variable ◽  
Dimensional Euclidean Space ◽  
Real Random Variable ◽  
Finite Dimensional

1. A type of problem which frequently occurs in probability theory and statistics can be formulated in the following way. We are given real-valued functions f(x), gi(x) (i = 1, 2, …, k) on a space (typically finite-dimensional Euclidean space). Then the problem is to set bounds for Ef(X), where X is a random variable taking values in , about which all we know is the values of Egi(X). For example, we might wish to set bounds for P(X > a), where X is a real random variable with some of its moments given.

Curve following in n—dimensional Euclidean space

SIMULATION ◽  
1973 ◽  
Vol 21 (5) ◽  
pp. 145-149 ◽  
Author(s):  
John Rees Jones
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space

scholarly journals On the Volume of the Zero Cell of a Class of Isotropic Poisson Hyperplane Tessellations

2014 ◽  
Vol 46 (3) ◽  
pp. 622-642 ◽  
Author(s):  
Julia Hörrmann ◽  
Daniel Hug
Keyword(s):  
Euclidean Space ◽  
Asymptotic Behaviour ◽  
Explicit Formula ◽  
Dimensional Euclidean Space ◽  
Arbitrary Dimensions ◽  
Zero Cell ◽  
Parametric Class

We study a parametric class of isotropic but not necessarily stationary Poisson hyperplane tessellations in n-dimensional Euclidean space. Our focus is on the volume of the zero cell, i.e. the cell containing the origin. As a main result, we obtain an explicit formula for the variance of the volume of the zero cell in arbitrary dimensions. From this formula we deduce the asymptotic behaviour of the volume of the zero cell as the dimension goes to ∞.

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